Rotate bones using a texture?
I don't know if it's possible, haven't found any mechanism to link texture value to bone transformation. But we can make random flailing movement by adding Noise modifier to the bone's rotation f-curve. From the manual entry on F-Curve Modifiers, this modifier

Modifies the curve with a noise formula. This is useful for creating
subtle or extreme randomness to animated movements, like camera shake.

If applied to the bone's rotation in at least one axis, it could create the effect you wanted: